A Lanczos bidiagonalization algorithm for Hankel matrices

A Lanczos bidiagonalization algorithm for Hankel matrices

This paper presents a fast algorithm for bidiagonalizing a Hankel matrix. An m×n Hankel matrix is reduced to a real bidiagonal matrix in O((m+ n)n log(m+ n)) floating-point operations (flops) using the Lanczos method with modified partial orthogonalization and reset schemes to improve its stability. Performance improvement is achieved by exploiting the Hankel structure, as fast Hankel matrix–ve...

The Lanczos Algorithm and Hankel Matrix Factorization

In 1950 Lanczos [22] proposed a method for computing the eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce the given matrix to a tridiagonal form, from which the eigenvalues could be determined. A characterization of the breakdowns in the Lanczos algorithm in terms of algebraic conditions of controllability and observability was addressed in [6] and [26]. Hankel matrice...

Lanczos Bidiagonalization With Partial Reorthogonalization

A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large Matrices

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A are almost linearly dependent. Furthermo...

A harmonic Lanczos bidiagonalization method for computing interior singular triplets of large matrices

This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted ha...